Both these devices act as sophisticated tone controls, in that they are used to alter specific frequencies in a signal, much as the bass, mid and high tone controls do on a home hi-fi.

The technical name for the circuits used in these devices are filters - they filter the signal and cut or boost certain portions of it. You may also see mention of high-pass and low-pass filters. These are simple filters that let only signals higher than a given frequency (a high-pass filter), or lower than a given frequency (a low-pass filter) through.
The specifications of these filters will also give some idea of the slope, or how steep the cut off is below the given frequency. Slopes of 12dB per octave are probably most useful in this context. Steeper slopes (up to 24dB/octave) can give phasing problems around the cut off point.
High-pass filters are commonly seen on mixer input stages, on crossovers and also on graphic eq units as a filter of, say 40Hz 12dB/octave is useful in reducing rumble and high power bumps (say if a mic is dropped) and doesn't affect the overall sound much as there is little audio information below this level.

A Graphic EQ is a multi-frequency filter, which generally can cut or boost at eight or more frequencies. It is important to remember that the frequencies that the filters are quoted to operate at are the centre frequencies. The filters will affect frequencies around them depending on how steep the slope is either side of this frequency. The steepness of the slope is defined by its Q value. Higher Q values give a denote a steeper slope.

High Q values (say 3 - 5) , despite being more selective, can give rise to 'ringing' or 'resonance', which is generally not desirable. Most graphic EQs have fixed Q values and fixed centre frequencies for each of their filters. See the Yamaha GQ2031 for an example of a typical Graphic EQ.